G11SPQ4 Lesson 1 Basic Concepts of Estimation 2024 03 11.pptx

Basic Concept of Estimation

1. MANUEL S. ENVERGA UNIVERSITY FOUNDATION BASIC EDUCATION DEPARTMENT Statistics and Probability Session 1 March 11, 2024 LUCENA CITY Facilitator: Mr. WILLIAM M. VERZO

2. MANUEL S. ENVERGA UNIVERSITY FOUNDATION BASIC EDUCATION DEPARTMENT Statistics and Probability Quarter 4 Lesson 1 Basic Concepts of Estimation LUCENA CITY Facilitator: Mr. WILLIAM M. VERZO

3. Lesson 1 Basic Concepts of Estimation Objectives: After completing this lesson, the learner must be able to: discuss the properties of a good estimator. illustrate point and interval estimation. distinguish between a point and an interval estimation. identify point estimation for the population mean and population variance. compute the point estimate of the population mean and the population variance.

4. Lesson 1 Basic Concepts of Estimation Introduction Inferential statistics focuses on estimating and predicting the results of a research study. Through the process of estimation, a parameter value is obtained using the information gathered from a particular sample. This process is not always accepted since interpretations and generalizations must be made from the data collected from the whole population.

5. Lesson 1 Basic Concepts of Estimation Introduction You have learned from previous lesson that a parameter is a numerical description of the entire population. Thus, parameters are usually unknown because it is infeasible to survey whole population. Estimation is the process used to calculate these population parameters by analyzing only a small random sample from the population. The value or range of values used to appropriate a parameter is called an estimate .

6. Lesson 1 Basic Concepts of Estimation Introduction The two types of parameter estimates are the point estimate and the interval estimate. Point estimate refers to a single value that best determines the true parameter value of the population. Interval estimate , on the other hand, gives a range of values within which the parameter value possibly falls. Suppose you work for a manufacturer of light bulbs and you want to predict the average life expectancy of all the light bulbs you produced.

7. Lesson 1 Basic Concepts of Estimation Introduction Using a sample of 100 light bulbs, you claim that the light bulbs last for an average of 5 months. This is a point estimate of the population parameter. However, if you state that the true life expectancy of the light bulbs is between 4 and 6 months, then you are giving an interval estimate of the parameter.

8. Lesson 1 Basic Concepts of Estimation Properties of a Good Estimator Sample measures, such as the sample mean, can be used to estimate population parameters, say the population mean. These sample measures are called estimators . The following are the properties of a good estimator: 1. Unbiasedness - Any parameter estimate can be considered a random variable since its value may change depending on certain factors including the selection of the members of the sample. Like all random variables, you can compute its expected value. An estimate is said to be unbiased

9. Lesson 1 Basic Concepts of Estimation Properties of a Good Estimator when the expectation (i.e., the mean) of all the estimates taken from samples with size n is shown to be equal to the parameter being estimate. 2. Consistency - The standard deviation of the sample statistics taken from the population is also the standard error. Thus, the standard deviation of an estimate is the standard error of that estimate and, thus, it gives the possible amount of error of predicting the population parameter. Consistency of an estimator is achieved when the estimate produced a relatively smaller standard error. This may be done by increasing the

10. Lesson 1 Basic Concepts of Estimation Properties of a Good Estimator sample used to estimate the population parameter. As the sample size increases, the value of the estimator approaches the value of the parameter being estimated. 3. Efficiency - From all the unbiased estimators of the population parameter, the efficient estimator is the one that gives the smallest variance.

11. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 1. Explain the goal of estimation and how it is carried out. The goal of approximation is to approximate the value of a parameter through a random sample analysis.

12. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 2. Differentiate point estimate from interval estimate. A point estimate determines the parameter using only a single value taken from the sample. An interval estimate gives not a specific value but a range of values in which the parameter falls.

13. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 3. Differentiate estimate from estimator. An estimator is a measurement taken from the sample, e.g., the sample mean. An estimate is a specific value or values of the estimator.

14. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 4. Enumerate the properties of a good estimator and describe how they may be achieved. The properties of a good estimator are unbiasedness, consistency, and efficiency . Unbiasedness is achieved when the mean of all the estimates computed from the sample is equal to the parameter that is estimated.

15. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 4. Enumerate the properties of a good estimator and describe how they may be achieved. Consistency is achieved when the standard deviation of all the estimates given by the estimator is kept smaller. Efficiency is achieved when the estimator gives the smallest variance among all the possible estimators.

16. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 5. Consider taking a point estimate from a randomly selected population of 120 students with a number of 75 high school students. Would you consider this a good estimate? Think of other factors that contribute to a good estimation. This could be a good estimate since the sample is randomly selected from the population and the sample is large. Sample size and randomization of the sample could be other factors that contribute to good estimation.

17. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 6. Under what condition is the sample mean an unbiased estimator of the population mean? The sample mean is an unbiased estimator of the population mean when the mean of all possible sample means is exactly the same as the true parameter.

18. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 7. You wish to estimate the average grade of high school students in your school from a random sample of 500 students. You may estimate the population parameter with either the mean or the median. Which you choose and why? The mean. Because the mean, although easily affected by extreme values, considers all the values in the sample. The median, on the other hand, ignores some of the values and changing them might not affect the median.

19. Lesson 1 Basic Concepts of Estimation Point Estimation As previously discussed, a point estimate of a population parameter is a sample statistic used to represent the true value of a parameter, and you endeavor to find the “best” point estimate for a given parameter. In the point estimation method, a population parameter is estimated simply with its corresponding sample statistic, i.e., the sample mean estimates the population mean, the sample variance estimates the population variance and so on.

20. Lesson 1 Basic Concepts of Estimation Point Estimation Frankly speaking, if X 1 , X 2 , X 3 , . . . X n denote the random variables from a random sample, then the function U (X 1 , X 2 , . . . X n ) used to estimate the population parameter q is called the point estimator of q . Specifically, the following functions are the point estimators of the population mean m , and the population variance s 2 , respectively.

21. Lesson 1 Basic Concepts of Estimation Point Estimation As you can see, the functions are the same formulas for sample mean and sample variance, respectively.

22. Lesson 1 Basic Concepts of Estimation Point Estimation Example 1: A researcher wants to estimate the average grade of all mathematics students in a certain school. He determined the grades of five students as follows: 76, 82, 88, 90, and 96. Estimate the average mathematics grade of all the students and the variance of their grades.

23. Lesson 1 Basic Concepts of Estimation Point Estimation Solution: The estimate of the population mean is 86.4. Now to get the estimate of the population variance,

24. Lesson 1 Basic Concepts of Estimation Point Estimation Solution:

25. Lesson 1 Basic Concepts of Estimation Interval Estimation A sample statistic contains all the information the sample provides. However, it is sometimes insufficient to estimate the value of the population parameter using only a single point estimator. For example, though the sample mean x is a “good” estimator of the population mean, m , x is not likely to be equal to m , and you are unsure of the accuracy of such a point estimate. But you can use your knowledge of the sampling distribution of x to construct an interval around the point estimate and to state your degree of certainty that the population mean m is

26. Lesson 1 Basic Concepts of Estimation Interval Estimation within that interval. Thus,, a < m < b, where a and b are the endpoints of the interval between which the population mean lies. This is the process of interval estimation and the interval of values that predicts where the true population parameter belongs is called interval estimate , or most commonly known as the confidence interval . You will observe that an interval estimate usually indicates accuracy where you can find the true sample estimate around the point estimate.

27. Lesson 1 Basic Concepts of Estimation Interval Estimation When this happens, it covers a specific range of interval within which the parameter of the population lies. You may estimate, for instance, the average salary of employees in a certain company but you are not sure how much an employee gets or the exact figure he or she gets. It is only within a given range of mean salaries of that company, which is confidential, and thus, you resort to guessing. You can only make estimation but not get the exact number.

28. Lesson 1 Basic Concepts of Estimation Interval Estimation The degree of certainty that the true population parameter falls within the constructed confidence interval is referred to as the confidence level . For example, if your interval estimate is made using an 80% confidence level, it indicates that your estimate is correct 80% of the time. Common choices for confidence levels are 90%, 95%, and 99%.

29. Lesson 1 Basic Concepts of Estimation BIG IDEA A good estimate relies on two factors: the confidence level and the length of the interval estimate. Shorter confidence interval are usually better and they can be obtained by increasing the same sample size or by using a sufficiently high confidence level.

30. Lesson 1 Basic Concepts of Estimation Interval Estimation Confidence levels correspond to probabilities (or percentages of the area) associated with the normal curve. To illustrate, a 90% confidence interval covers 90% of the normal curve. Since the curve is symmetric, then the interval covers 45% of the area to the right and 45% to the left, and the probability of observing a value outside this interval, usually denoted as a , is 10%: 5% to the right and 5% to the left.

31. Lesson 1 Basic Concepts of Estimation Interval Estimation The figure below illustrates a 90% confidence interval on the standard normal curve.

32. Lesson 1 Basic Concepts of Estimation Interval Estimation In general, if C represent the confidence level of the interval estimate and a the area outside the boundaries of the interval estimate, then under the standard normal curve, the interval estimate may be written as – z a / 2 < Z < z a / 2 , where z a / 2 is simply the z–score corresponding to the probability C / 2 found in the z–distribution table.

33. Lesson 1 Basic Concepts of Estimation Interval Estimation Note that a = 1 – C is the area on the tails under the normal curve. One may use the alpha in locating the z–value in constructing the confidence interval in the last figure. Further, remember that because the standard normal table is based on areas between z = 0 and z a / 2 , the z–value is found by locating the area of 0.5 – a / 2 which is the part of the normal curve between the middle of the curve and one of the tails. Or you may locate this z–value by changing the confidence level from percentage to proportion, divide it in half,

34. Lesson 1 Basic Concepts of Estimation Interval Estimation Or go to the table with this value. Study the following examples.

35. Lesson 1 Basic Concepts of Estimation Interval Estimation Example 2 Find an interval of values for Z using the standard normal distribution, corresponding to an area of 95%. Solution: Given C = 0.95, C / 2 = 0.475

36. Lesson 1 Basic Concepts of Estimation Interval Estimation The value of z corresponding to the area 0.475 is 1.96. Thus, the 95% confidence interval is –1.96 < Z < 1.96. Using interval notation this may be written as (–1.96, 1.96).

37. Lesson 1 Basic Concepts of Estimation Interval Estimation Example 3 Find an interval of values for Z using the standard normal distribution, corresponding to an area of 90.1%. Solution: Given C = 0.901, C / 2 = 0.4505 and z a / 2 = +/- 1.65 The 90.1% confidence interval is –1.65 < Z < 1.65. or (–1.65, 1.65).

38. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 1. Differentiate the processes of point estimation and interval estimation. Point estimation involves using a single-value estimate while interval estimation constructs an interval with a certain confidence level around which the parameter lies.

39. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. 2. What is a confidence interval and how does the choice of confidence level affect the interval estimation? Discuss. Confidence interval is an interval estimate of the population parameter based on a certain degree of confidence or certainty. The choice of the confidence level affects the interval estimation because a different confidence level changes the values of the lower limit and the upper limit of the interval estimate.

40. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. A shoe manufacturer wants to estimate the average shoe size of the Filipino teens. Using a sample of 7 random teenagers, he gathered the following sizes: 38, 42, 43, 43, 44, 45, and 47. a. Which point estimator will be most appropriate to estimate the population parameter? The sample mean is the appropriate point estimator of the parameter because the average shoe size is being estimated in the context of the problem.

41. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. A shoe manufacturer wants to estimate the average shoe size of the Filipino teens. Using a sample of 7 random teenagers, he gathered the following sizes: 38, 42, 43, 43, 44, 45, and 47. b. Estimate the average shoe size of Filipino teens using the sample. Is this a reasonable estimate? 43.14 Yes

42. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. A shoe manufacturer wants to estimate the average shoe size of the Filipino teens. Using a sample of 7 random teenagers, he gathered the following sizes: 38, 42, 43, 43, 44, 45, and 47. c. Estimate the variance of the shoe sizes of Filipino teens. 7.81

43. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. For C = 80%, 90%, 95% and 99%, find z so that P (– z < Z < z) = C. 80% confidence level → 1.282 90% confidence level → 1.645 95% confidence level → 1.96 99% confidence level → 2.575

44. Lesson 1 Basic Concepts of Estimation Classwork Read and analyze the following questions. Find an interval of values for Z so that P (– z < Z < z) = 0.8836. – 1.57 < Z < 1.57

45. Lesson 1 Basic Concepts of Estimation Practice Activity 1: Divide using long division. 1. x 3 + 3 x 2 – 2 x + 1) ÷ (x + 1) a 20 = 80

46. Lesson 1 Basic Concepts of Estimation Enrichment Activity 1: Guess what's Next? The illustrations above illustrate a sequence involving

47. MANUEL S. ENVERGA UNIVERSITY FOUNDATION BASIC EDUCATION DEPARTMENT Please Do! Lesson 1 Homework 1 LUCENA CITY Facilitator: Mr. WILLIAM M. VERZO

48. Lesson 1 Basic Concepts of Estimation

49. Lesson 1 Basic Concepts of Estimation Basic Concept of Estimation Properties of a Good Estimator Point Estimation Interval Estimation

50. Lesson 2 Estimating Population Means Estimating Population Means Estimating Population Mean Using A Large Sample: Applying the Central Limit Theorem Estimating Population Mean with Unknown Variance Using A Small Sample The t-Distribution Distribution Tables at Different Levels Estimating Population Means Usinf t-Distribution

51. Lesson 3 Estimating Population Proportion The Sample Properties as a Point Estimator Distribution of Sample Proportions Probabilities Involving Distribution of Sample Proportions Estimating the Population Proportion Confidence Interval to Estimate the Difference Between Two Population Proportions